Yes, and no. The budget is a planning document initially, but it is the basis of the appropriations bills that determine government expenditure on items that are not already authorized. Therefore, the budget strongly influences flows in the period and beyond the period in some cases where spending continues.

The key point is that the budget does not determine government spending owing to the fact that 1) previous law has mandated spending across periods, e.g., SS, etc., and also automatic stabilization that operates counter-cyclically to offset contractions. In addition, there can be off-budget appropriations.

So while it is incorrect to equate "the budget" with the total of government spending in a period (G), the budget does play an important role.

But the same terms can have different meaning in different contexts, and "budget" is one of the terms that is used loosely.

Confusion can result when ordinary language use is confused with technical definition in accounting.